ANISOTHTES SOBOLEV KAI EFARMOGES

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1 NIKOLAOS K. TABOULARHS MAJHMATIKOS ANISOTHTES SOBOLEV KAI EFARMOGES Didaktorik Diatrib PANEPISTHMIO PATRWN TMHMA MAJHMATIKWN PATRA 004

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3 Ston Patèra mou, me eugnwmosônh 3

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5 5 EuqaristÐec H Didaktorik aut diatrib ekpon jhke sto Tm ma Majhmatik n tou PanepisthmÐou Patr n, me epiblèponta ton Kajhght tou Tm matoc Majhmatik n k. Ajanˆsio Kotsi lh kai mèlh thc sumbouleutik c epitrop c ton Kajhght tou Tm matoc Qhmik n Mhqanik n tou PanepisthmÐou Patr n k. Ge rgio Dˆsio kai ton EpÐkouro Kajhght tou Tm matoc Majhmatik n k. Dhm trio Hliìpoulo. Jewr qrèoc mou na ekfrˆsw tic eilikrineðc kai egkˆrdiec euqaristðec mou ston Kajhght k. Ajanˆsio Kotsi lh gia thn upìdeixh tou jèmatoc aut c thc diatrib c, gia to suneqèc endiafèron tou kaj ìlh th diˆrkeia ekpìnhshc thc kai thn kajoristik sumbol tou sthn olokl rwsh aut c. Jermˆ euqarist ton Kajhght k. Ge rgio Dˆsio kai ton EpÐkouro Kajhght k. Dhm trio Hliìpoulo gia th sumbol touc kai gia tic parathr seic touc sto telikì keðmeno thc diatrib c. EpÐshc,euqarist touc Kajhghtèc Thierry Aubin tou PanepisthmÐou Université de Paris IV GallÐa) kai Ge rgio Alexìpoulo tou PanepisthmÐou Université Paris-Sud, Orsay GallÐa) gia thn episthmonik sunergasða kai tic qr simec episthmonikèc suzht seic pou eðqa mazð touc. Katˆ thn diˆrkeia thc ekpìnhshc aut c thc diatrib c eðqa epitôqei katìpin exetˆsewn metaptuqiak c upotrofðac pou mou pareðqe to 'Idruma Kratik n Upotrofi n I.K.U.), to opoðo kai euqarist. Tèloc aisjˆnomai bajeiˆ yuqik anˆgkh na ekfrˆsw thn eugnwmosônh mou ston Patèra mou pou eulabikˆ afosi jhke sthn prìodo mou kai se ìti kalì gia mèna sth zw. EpÐshc ja jela na anaferj me sugkðnhsh sthn Mhtèra mou AikaterÐnh kai sthn Giagiˆ mou Alexˆndra, mˆna tou Patèra mou allˆ kai pou kai eg

6 6 thn jewroôsa san kai dik mou afoô tìso polô me agˆphse kai me frìntise - tan gia mèna pragmatik eulogða sth zw mou - kai pou h fug touc apì thn ed, thn g inh zw, me lôphse bajeiˆ yuqikˆ. Nikìlaoc K. Taboulˆrhc Majhmatikìc Panepist mio Patr n Tm ma Majhmatik n

7 Perieqìmena 1 Eisagwg Istorik Anadrom tou jèmatoc Perilhptik parousðash thc diatrib c Basikèc ènnoiec 1.1 Oloklhrwtikèc Anisìthtec O Metasqhmatismìc Fourier Q roi Sobolev Oi q roi H s ) kai H s,p ) O q roc H s ) Idiìthtec tou q rou H s ) O q roc H s,p ) Bèltistec stajerèc gia anisìthtec Sobolev me an terhc mh akèraiac tˆxhc parag gouc Anisìthtec gia telestèc Riesz Anisìthtec gia telestèc Bessel Anisìthtec gia ton telest tou Yukawa MÐa Logarijmik Anisìthta Sobolev MÐa Logarijmik Anisìthta Sobolev Efarmogèc pˆnw sthn sfaðra S n Telestèc pˆnw sthn sfaðra S n Intertwining operators)

8 8 PERIEQŸOMENA 6. Anisìthtec pˆnw sthn sfaðra S n 'Allec efarmogèc twn Anisot twn Asjen c sôgklish 'Ena Prìblhma Metabol n Variational Problem) 111 BibliografÐa 17

9 Kefˆlaio 1 Eisagwg 1.1 Istorik Anadrom tou jèmatoc Se genikèc grammèc, mða anisìthta Sobolev èqei na kˆnei me mða ektðmhsh twn kˆtw tˆxhc parag gwn miac sunˆrthshc se ìrouc twn an terhc tˆxhc parag gwn. Tètoiec ektim seic, pou isqôoun gia sunart seic se sugkekrimènec klˆseic, èqoun gðnei èna basikì ergaleðo sthn Ôparxh kai kanonikìthta jewri n gia lôseic merik n diaforik n exis sewn, sto logismì twn metabol n, sth gewmetrik jewrða mètrou kai se ˆllouc klˆdouc thc Anˆlushc. Oi arqikèc idèec emfanðsjhkan to 1930 me thn ergasða tou G. A. Bliss, An integral inequality pou dhmosieôjhke sto periodikì Journal of London Mathematical Society, [15]). Sthn ergasða aut parousiˆzetai mia endiafèrousa oloklhrwtik anisìthta pou eðqe tejeð apì touc G. H. Hardy kai J. E. Littlewood [46]) sthn mða diˆstash kaj c epðshc kai ekeðnec oi sunart seic oi opoðec dðnoun thn isìthta sthn anisìthta. H pragmatik spoudaiìthta twn jemˆtwn aut n dìjhke lðga qrìnia argìtera me tic ergasðec tou S. L. Sobolev. O S. L. Sobolev to 1938 me thn ergasða tou On a theorem of functional analysis, pou dhmosieôjhke sto periodikì Mat. Sb ) kai metafrˆsthke argìtera apì to periodikì thc Amerikˆnikhc Majhmatik c EtairÐac Amer. Math. Soc. Translations eis gage entel c nèo trìpo antimet pishc problhmˆtwn thc Majhmati- 9

10 10 KEFŸALAIO 1. EISAGWGŸH k c Fusik c kai ìqi mìnon) pou anˆgontai se Grammikèc kai Mh Grammikèc Merikèc Diaforikèc Exis seic. 'Ektote, up rxe almat dhc anˆptuxh nèwn Mejìdwn epðlushc aut n twn Exis sewn. 'Etsi loipìn genn jhkan nèoi sunarthsiakoð q roi lôsewn aut n twn Exis sewn pou onomˆzontai s mera Q roi Sobolev. O S. L. Sobolev apèdeixe genikèc oloklhrwtikèc anisìthtec gia diaforðsimec sunart seic poll n metablht n kai eisˆgontac thn ènnoia thc genikeumènhc parag gou je rhse touc q rouc Banach W m,p Ω) twn sunart sewn ston q ro L p Ω), p 1, me genikeumènec parag gouc tˆxhc m oi opoðec eðnai oloklhr simec tˆxhc p. Eidikìtera, qrhsimopoi ntac ta jewr matˆ tou pˆnw se dunamikoô tôpou oloklhr mata potential type integrals) kaj c epðshc kai thn oloklhrwtik anaparˆstash twn sunart sewn apèdeixe thn emfôteush twn q rwn W m,p Ω) ston L q Ω) ston q ro twn suneq n sunart sewn CΩ) kˆtw apì sugkekrimènec sunj kec gia touc ekjètec m, p, q. To epanastatikì je rhma tou Sobolev eðnai h gnwst plèon klassik anisìthta Sobolev: fx) n/n ) dx se pio sôntomh èkfrash ) n n C fx) dx ) 1 1.1) f q C f, q = n n pou isqôei gia kˆje diaforðsimh sunˆrthsh f me sumpag forèa, ìpou f sumbolðzei to gradient thc f kai h stajerˆ onomazìmenh plèon stajerˆ Sobolev C > 0) exartˆtai mìno apì thn diˆstash n >. O ekjèthc q = n n eðnai o monadikìc ekjèthc gia ton opoðo mia tètoia anisìthta mporeð na isqôei kai gi' autì lègetai krðsimoc arijmìc thc anisìthtac 1.1). Prˆgmati kˆtw apì tic diastolèc dilations) tou,

11 1.1. ISTORIKŸH ANADROMŸH TOU JŸEMATOS 11 x λx και fx) fx/λ) o telest c pollaplasiˆzetai me λ 1, en ta n-diˆstata oloklhr mata me λ n. 'Etsi, to aristerì mèroc thc 1.1) pollˆplasiˆzetai me λ n/q en to dexiì mèroc thc eðnai anˆlogo tou λ n. Profan c, ta dôo mèrh mporoôn na sugkrijoôn ìtan autˆ eðnai bajmwtˆ ìmoia, kˆti to opoðo odhgeð sto ìti q = n/n ). 'Ena ˆllo jèma pou prèpei na shmei soume eðnai to ìti h 1.1) isqôei mìno gia n > kai wc ek toôtou tðjetai to er thma gia to poiec anisìthtec ja èprepe na antikatast soun thn 1.1) se mða kai dôo diastˆseic. Upˆrqoun pollèc diaforetikèc apant seic, h pio sunhjismènh apì autèc eðnai [61]) f q S,q f + f ) για κάθε q < για n = kai f df S 1,q dx + f ) για n = 1 ìpou oi bèltistec stajerèc S,q kai S 1,q na èqoun antðsoiqa ta ˆnw frˆgmata kai ) 1 /q q S,q < q 1) +/q q 4/q 8π ) 1 /q q S 1,q < q 1) 1+1/q q 1 /q. π Argìtera ta jewr mata tou Sobolev genikeôjhkan katˆ poikðllouc trìpouc Kondrashov [56]), Il in [51]), Gagliardo [41]), Nirenberg [75]), k.a.). Se autèc tic melètec tì pedðo orismoô

12 1 KEFŸALAIO 1. EISAGWGŸH twn sunart sewn jewreðtai na èqei thn legìmenh kwnik idiìthta kˆje shmeðo tou pedðou orismoô eðnai h koruf enìc sfairikoô k nou me stajerì Ôyoc kai gwnða kai o opoðoc brðsketai mèsa se autì). AxÐzei na shmei soume to ìti oloklhrwtikèc anisìthtec isqôoun kˆtw apì asjeneðc apait seic gia to pedðo orismoô twn sunart sewn tan gnwstì akìma kai prin apì ta apotelèsmata tou Sobolev. Gia parˆdeigma h anisìthta tou Friedrich [40]) ) f dx K f) dx + f ds Ω Ω Ω eðnai alhj c kˆtw apì thn monadik upìjesh ìti Ω eðnai mða fragmènh perioq gia thn opoða isqôei o tôpoc Gauss-Green. To 1933 o Nikodým [74]) èdwse èna parˆdeigma miac perioq c Ω tètoiac ste h tetragwnik oloklhrwsimìthta thc gradient den sunepˆgetai thn tetragwnik oloklhrwsimìthta miac sunˆrthshc orismènhc sthn perioq Ω. H monografða twn Courant kai Hilbert [3]) perièqei ikanèc sunj kec gia thn isqô thc anisìthtac tou Poincaré f dx K f) dx + 1 ) fdx Ω Ω Ω O gnwstìc, apì to brabeðo Nobel sthn oikonomða J. Nash [73]) apèdeixe thn anisìthta Nash f 1+/n) C f f 4/n 1 1.) pou eðnai isodônamh me thn anisìthta 1.1) kai gi' autì lègetai kai anisìthta tôpou Sobolev. EpÐshc h anisìthta Nash 1.) eðnai isodônamh me thn sqèsh ektðmhsh jermoô pur na ht, x.y)) sup x,y ht, x, y) Ct n/ t > 0 ìpou ht, x, y) sumbolðzei thn jemeli dh lôsh tou probl matoc Cauchy Ω

13 1.1. ISTORIKŸH ANADROMŸH TOU JŸEMATOS 13 kai { u t = u u0, x) = δx y) ) 1 x y ht, x, y) = 4πt) n/exp 4t ìpou u = ut, x) orðzetai sto 0, + ), y, δ h sunˆrthsh Dirac kai o telest c tou Laplace bl. L. Saloff - Coste, Aspects of Sobolev - Type Inequalities, Cambridge University Press, 00). Pr th genðkeush thc 1.1) eðnai h bèltisth anisìthta Sobolev fx) np/n p) dx ) n p np ) 1 Cn, p) fx) p p dx 1.3) ìpou 1 p < n. O arijmìc np n p eðnai epðshc krðsimoc arijmìc thc anisìthtac 1.3). H akrib c tim thc stajerˆc Sobolev Cn, p) sthn 1.3) upologðsthke to 1976 apì touc Th. Aubin [3]) kai G. Talenti [87]) anexˆrthta o ènac apì ton ˆllo) en h akrib c tim thc stajerˆc Nash sthn 1.) upologðsthke to 1993 apì touc Carlen kai Loss [5]). An p = 1 tìte h 1.3) grˆfetai f n/n 1 Cn, 1) f 1 kai eðnai isodônamh ergasða twn Federer and Fleming, [38]) me thn isoperimetrik anisìthta Ω Ω n 1)/n 1 Cn, 1) ìpou Ω eðnai èna leðo fragmèno anoiktì uposônolo tou, Ω sumbolðzei ton ìgko tou Ω kai Ω ton ìgko tou sunìrou Ω tou Ω.

14 14 KEFŸALAIO 1. EISAGWGŸH EpÐshc an p = n tìte sthn jèsh thc anisìthtac 1.3) èqoume thn anisìthta twn Trudinger-Moser [71]) Ω ] exp [α fx) n/n 1) dx Cn) Ω ìpou Ω, Ω ìpwc prin, kai α nω 1/n 1) n 1. To ω n 1 sumbolðzei ton ìgko thc monadiaðac sfaðrac S n 1 diˆstashc n 1. Er thma: MporoÔme na genikeôsoume kai ˆllo thn anisìthta 1.3) kai proc poia katèujunsh ; 1. AntÐ gia Ω mporoôme na èqoume mða pollaplìthta M;. IsqÔei h 1.3) gia an terhc tˆxhc parag gish ; O Th. Aubin tan apì touc pr touc pou asqol jhke me thn anˆlush se pollaplìthtec Riemann kai eis gage tic anisìthtec Sobolev se sumpageðc kai mh sumpageðc pl reic) pollaplìthtec Riemann bl. Th. Aubin Some Nonlinear Problems in Riemannian Geometry Springer - Verlag 1998 kai E. Hebey majht c tou Aubin) Nonlinear Analysis on Manifolds : Sobolev Spaces and Inequalities Courant Lecture Notes, American Mathematical Society, 000). 'Etsi an M, g) eðnai mia pl rhc pollaplìthta Riemann M me metrik g kai h M eðnai fragmènhc kampulìthtac, oi Hebey - Vaugon apèdeixan to 1995 [49]) thn bèltisth anisìthta Sobolev: M ) n f n n n dvg) C n, ) f dvg) + B M M f dvg) 1.4) ìpou ektìc apì thn bèltisth stajerˆ Sobolev Cn, ) emfanðzetai kai mia nèa stajerˆ B pou exartˆtai apì to n > kai to frˆgma thc kampulìthtac thc M. Genikìtera gia 1 p < n h parakˆtw bèltisth anisìthta Sobolev stic Pollaplìthtec Riemann apodeðqjhke to 1999 anexˆrthta apì ton Druet kai touc

15 1.1. ISTORIKŸH ANADROMŸH TOU JŸEMATOS 15 Aubin - Li bl. Hebey - Druet The AB Program in Geometric Analysis : Sharp Sobolev Inequalities and Related Problems Memoirs of the American Mathematical Society 00) M ) n p f pn pn n p dvg) Cn, p) +B M M ) 1 f p p dvg) f p dvg) ) 1 p 1.5) An h pollaplìthta M eðnai h monadiaða sfaðra S n +1, oi Th. Aubin - A. Cotsiolis bl. Equations elliptiques non linéaires sur S n dans le cas supercritique Bull. Sci. Math ), ) axiopoi ntac tic summetrðec thc S n upolìgisan thn bèltisth stajerˆ Sobolev Cn, k) gia thn sfaðra S n sthn anisìthta ìpou S n u k/k ) dv k k > n. n ) k k Cn, k) u dv + B u dv S n S n 1.6) Gi' autì o arijmìc k/k ) lègetai uperkrðsimoc arijmìc thc bèltisthc Anisìthtac Sobolev. EpÐshc upolìgisan thn bèltisth stajerˆ ν sthn anisìthta Trudinger - Moser S 3 e ϕ dv C ν exp [ ] ν ϕ dv + S 3 ϕdv/ω 3 S 3 1.7) ìpou ω 3 eðnai o ìgkoc thc sfaðrac S 3. Oi sunart seic u eðnai analloðwtec apì thn drˆsh thc omˆdac Ok) Om) me k + m = n + 1, k m ) upoomˆdac twn orjogwnðwn metasqhmatism n On+1) kai oi sunart seic ϕ apì thn drˆsh thc omˆdac

16 16 KEFŸALAIO 1. EISAGWGŸH O) O). En suneqeða efarmìzontac ta prohgoômena èlusan antðstoiqa dôo exis seic me uperkrðsimouc ekjètec : 1) u + αx)u = fx)u k+)/k ) ) ϕ + αx) = fx)e ϕ ìpou oi sunart seic suntelest n αx), fx) epðshc eðnai analloðwtec apì tic antðstoiqec omˆdec. Parˆllhla o D. Iliopoulos bl. Best constants for supercritical Sobolev inequalities and application to some quasilinear supercritical elliptic equation of scalar curvature type Bull. Sci. Math. 14, 000) ) axiopoi ntac epðshc tic summetrðec thc S n upolìgise thn bèltisth stajerˆ Sobolev Ck, q, m) gia thn sfaðra S n sthn anisìthta S n u kp/k p) dv ) k p kp ) 1/p Ck, p, m) u p dv S n ) 1/p +B u dv 1.8) S n ìpou kp k p > np n p k + m = n + 1). O arijmìc kp/k p) eðnai uperkrðsimoc arijmìc thc Anisìthtac Sobolev 1.8). EpÐshc èluse thn wc ˆnw exðswsh 1 me anˆlogec sunj kec. Oi Th. Aubin - A. Cotsiolis bl. Equations non linéaires avec le p-laplacien et la fonction exponentielle sur les variétés Riemanniennes compactes Bull. Sci. Math. 14, 1 000), 1-19 ): 1. 'Elusan thn exðswsh n ψ+α = fx)e ψ pˆnw se mia tuqoôsa pollaplìthta Riemann M diˆstashc n > 1, ìpou n ψ = div ψ n ψ ) kai α R, f plhreð kˆpoiec sunj kec.. Pˆnw sthn sfaðra S n n > : i) axiopoi ntac thn bèltisth anisìthta Trudinger - Moser

17 1.1. ISTORIKŸH ANADROMŸH TOU JŸEMATOS 17 e ϕ C ν exp S n èlusan thn exðswsh [ ] [ν ϕ dv + S n ϕdv/ω n S n k ψ + α = fx)e ψ ii) axiopoi ntac thn bèltisth anisìthta Sobolev 1.8) gia thn S n èlusan thn exðswsh p u + αx)u p 1 = fx)u p 1 ìpou p = kp k p > np n p, m + k = n + 1, k m kai p eðnai uperkrðsimoc arijmìc thc bèltisthc Anisìthtac Sobolev gia thn S n. Apì ta teleutaða gðnetai katafan c h pragmatik spoudaiìthta twn anisot twn Sobolev sthn lôsh Merik n Diaforik n Exis sewn.

18 18 KEFŸALAIO 1. EISAGWGŸH 1. Perilhptik parousðash thc diatrib c H paroôsa diatrib entˆssetai ereunhtikˆ sthn perioq thc Mh Grammik c Anˆlushc kai eidikìtera sthn eôresh bèltistwn stajer n gia anisìthtec Sobolev ston q ro me an terhc tˆxhc dekadikèc parag gouc. EpÐshc, dðnontai oi antðstoiqec bèltistec stajerèc aut n twn anisot twn pˆnw sthn sfaðra S n me thn qrhsimopoðhsh wc basikoô ergaleðou thn stereografik probol. Tèloc san mða efarmog twn eurejèntwn anisot twn èqoume èna je rhma sqetikì me autì twn Rellich-Kondrashov kai to opoðo eðnai exairetik c shmasðac, idiaðtera ston logismì twn metabol n. Sto Kefˆlaio 1 1.1) ègine mia sôntomh istorik anadrom sta jèmata pou anafèrontai stic Anisìthtec Sobolev kai sthn exeliktik touc poreða apì tic arqèc tou perasmènou sqedìn ai na mèqri s mera. Paratèjhkan ta pio shmantikˆ mèqri stigm c ereunhtikˆ apotelèsmata pˆnw sto jèma twn Anisot twn Sobolev kai dìjhkan mða seirˆ apì sqetikˆ probl mata pou lônontai qrhsimopoi ntac autèc peremfereðc me autèc aniosìthtec. Sto Kefˆlaio dðnontai gia lìgouc plhrìthtac kai saf neiac kˆpoiec gnwstèc kai polô qr simec oloklhrwtikèc anisìthtec kai kˆpoia basikˆ jèmata thc Anˆlushc ìpwc o Metasqhmatismìc Fourier, oi Katanomèc kai oi q roi Sobolev. Sto Kefˆlaio 3 arqikˆ paratðjentai oi orismoð twn telest n Riesz ) s/ ), Bessel I ) s/ ) kai Yukava + µ ). Oi q roi H s ), me s kˆje pragmatikì arijmì, perièqoun tic f L )-sunart seic gia tic opoðec to ) s/ f eðnai epðshc mða L )-sunˆrthsh. AutoÐ oi q roi onomˆzontai epðshc q roi Sobolev twn sunart sewn me dekadik c tˆxhc parag gouc. ApodeiknÔoume thn isodunamða twn norm n twn q rwn H s ) kai W s, ) ìtan s eðnai ènac jetikìc akèraioc kai to Je rhma 3..6 [8]) bebai nei thn puknìthta tou q rou C c ) stouc q rouc H s ). Sto Kefˆlaio 4 paratðjentai ta kôria jèmata aut c thc dia-

19 1.. PERILHPTIKŸH PAROUSŸIASH THS DIATRIBŸHS 19 trib c ta opoða en suntomða anafèroume sto shmeðo autì. MÐa deôterh genðkeush thc Anisìthtac 1.1) eðnai h akìloujh bèltisth Anisìthta Sobolev an terhc aujaðrethc - dhlad akìma kai dekadik c - tˆxhc parag gishc f n/n s) C )s/ f 1.9) ìpou s eðnai ènac jetikìc pragmatikìc arijmìc me n > s. An s = 1 tìte h 1.9) sumpðptei me thn 1.1). To apotèlesma autì eðnai prwtìtupo kai dhmosieôjhke to 00 sthn ergasða me tðtlo Sharp Sobolev type inequalities for higher fractional derivatives [8]). EpÐshc, sth sunèqeia genikeôoume to parapˆnw apotèlesma jewr ntac ìti ) s/ f eðnai mða sunˆrthsh tou q rou L p ) lambˆnontac tic akìloujec bèltistec Anisìthtec f q K ) s/ f p 1.10) gia sugkekrimèna p kai q sthn kˆje perðptwsh kai dðnoume èna ˆnw frˆgma thc bèltisthc stajerˆc gia tuqaða p kai q. Akìmh brðskoume tic bèltistec stajerèc stic anisìthtec kai f q B I ) s f 1.11) f q V + µ )f 1.1) pou perièqoun telestèc Bessel kai Yukawa antðstoiqa. Ta apotelèsmata autˆ pou èqoun gðnei dektˆ gia dhmosðeush brðskontai sthn ergasða me tðtlo Sharp inequalities for Riesz, Bessel and Yukawa potential operators [30]). Tèloc gia to R kai to R [8],[9],[30]) dðnoume ˆnw frˆgmata gia tic bèltistec stajerèc twn anisot twn: f q S [ ] ) s/ f + f, f H s R), 1.13)

20 0 KEFŸALAIO 1. EISAGWGŸH kai f q B I ) s f, f H s R ), 1.14) f q V + µ )f, f H 1 R). 1.15) Sto Kefˆlaio 5 parousiˆzoume mða bèltisth logarijmik anisìthta Sobolev gia an terhc tˆxhc parag gouc dhlad dðnoume mða nèa logarijmik anisìthta pou perièqei ton telest ) s/ [31]). Sto Kefˆlaio 6 dðnoume Anisìthtec pˆnw sthn sfaðra S n jewr ntac sômmorfa analloðwtouc diaforikoôc telestèc an terhc tˆxhc se aut n pou ousiastikˆ eðnai h metaforˆ sthn sfaðra mèsw thc stereografik c probol c twn yeudodiaforik n telest n ) s/ tou [9], [30]). Sto Kefˆlaio 7 dðnoume kˆpoiec efarmogèc twn apotelesmˆtwn pou d same stic prohgoômenec paragrˆfouc. To pr to apotèlesma pou parajètoume sthn parˆgrafo aut, efarmìzontac thn Anisìthtac 1.9), eðnai mða nèa ekdoq tou gnwstoô jewr matoc twn Rellich-Kondrashov pou kontolog c anafèrei ìti h asjen c sôgklish miac akoloujðac ) s/ f j sunepˆgetai thn isqur sôgklish thc f j ston L p ) pˆnw sta mikrˆ sônola peperasmènou mètrou) kai gia sugkekrimèna p [9]). EpÐshc, qrhsimopoi ntac autì to apotèlesma, lônoume èna prìblhma metabol n variational problem) to opoðo sunðstatai sto na elaqistopoi - soume to akìloujo sunarthsiakì EΨ) = T Ψ + V Ψ me thn sunj kh ìti Ψ = 1 kai ìpou T Ψ = ) s/ Ψ και V Ψ = V x) Ψx) dx.

21 Kefˆlaio Basikèc ènnoiec Sto kefˆlaio autì dðnontai gia lìgouc plhrìthtac kai saf - neiac kˆpoiec gnwstèc kai polô qr simec oloklhrwtikèc anisìthtec kai kˆpoia basikˆ jèmata thc Anˆlushc ìpwc o Metasqhmatismìc Fourier, oi Katanomèc kai oi q roi Sobolev..1 Oloklhrwtikèc Anisìthtec Oi anisìthtec oi opoðec ja dojoôn sthn parˆgrafo aut eðnai perðplokec kai den prokôptoun apì tic idiìthtec thc kurtìthtac. H EukleÐdeia dom tou paðzei ed èna shmantikì rìlo. O prosdiorismìc twn bèltistwn stajer n sharp or optimal constants) kai oi peript seic thc isìthtac eðnai to dôskolo prìblhma. Den eðnai dôskolo na parˆgoume autèc tic anisìthtec an den apaitoôntai bèltistec stajerèc, an kai ja prèpei na eipwjeð gia lìgouc istorik c akrðbeiac ìti akìma kai h mh bèltisth ekdoq thc anisìthtac Hardy-Littlewood-Sobolev den tan kai tìso eôkolo na apodeiqjeð. Den eðnai profanèc oôte akìma aparaðthta alhjèc se ìlec tic peript seic, ìti oi bèltistec morfèc aut n twn anisot twn mporoôn na epiteuqjoôn apì sugkekrimènec sunart - seic. Autèc oi sunart seic gia tic opoðec h anisìthta gðnetai isìthta, onomˆzontai megistopoioôsec maximizers) akìma elaqistopoioôsec minimizers). O ìroc beltistopoioôsec optimizer) epðshc qrhsimopoieðtai suqnˆ. 1

22 KEFŸALAIO. BASIKŸES ŸENNOIES MÐa sunˆrthsh Gauss Gaussian function) g : C eðnai mða fragmènh sunˆrthsh thc morf c gx) = exp{ x, Ax) + ix, Bx) + J, x) + C}, ìpou A kai B eðnai pragmatikoð summetrikoð pðnakec me A jetikˆhmiorismènoc dhl. x, Ax) 0 gia ìla ta x ) kai me J C n. An g L p ) gia kˆpoio p <, tìte A prèpei na eðnai jetikˆ orismènoc. UpeujumÐzoume epðshc ìti f g sumbolðzei thn sunèlixh twn sunart sewn f kai g. Pr ta ja anafèroume thn Anisìthta Young. Je rhma.1.1. [61]) Anisìthta Young) 'Estw p, q, r 1 kai 1/p + 1/q + 1/r =. 'Estw f L p ), g L q ) kai h L r ). Tìte fx)g h)x)dx = fx)gx y)hy)dxdy C p,q,r;n f p g q h r..1) H bèltisth stajerˆ C p,q,r;n isoôtai me C p C q C r ) n C p = p 1/p /p 1/p..) kai ìpou 1/p + 1/p = 1 An p, q, r > 1, tìte h anisìthta.1) gðnetai isìthta an kai mìnon an f, g kai h eðnai oi sunart seic Gauss fx) = Aexp[ p x α, Jx α)) + ik x], gx) = Bexp[ q x b, Jx b)) + ik x],.3) hx) = Cexp[ r x c, Jx c)) + ik x], ìpou A, B, C C ; α, b, c, k me α = b + c kai J eðnai kˆje pragmatikìc, summetrikìc, jetikˆ-orismènoc pðnakac.

23 .1. OLOKLHRWTIKŸES ANISŸOTHTES 3 Parathr seic : 1) C p = 1/C p. ) Qrhsimopoi ntac thn anisìthta Hölder, eðnai eôkolo na doôme ìti ìtan g kai h dðnontai, h kalôterh epilog gia thn f eðnai fx) = e iθx) g h)x) p /p, ìpou θx) orðzetai apì thn sqèsh g h = e iθ g h. 'Etsi, h anisìthta Young mporeð na grafeð : g h p C q C r /C p ) n g q h r = C p,q,r;n g q h r me 1/q + 1/r = 1 + 1/p. 3) H bèltisth stajerˆ brèjhke tautìgqrona apì touc Beckner [10]) kai Brascamp-Lieb [17]). H sunj kh gia thn isìthta dìjhke argìtera. 4) Pl rwc genikeumènh anisìthta Young. 'Estw k > 1, akèraioi n 1,..., n k kai arijmoð p 1,..., p k > 1. 'Estw M 1 kai èstw B i gia i = 1,..., k) na eðnai mða grammik apeikìnish apì to R M sto i. 'Estw Z : R M R + na eðnai kˆpoia sugkekrimènh sunˆrthsh Gauss, Zx) = exp{ x, Jx)} me J ènac pragmatikìc, jetikˆ-hmiorismènoc M M pðnakac pijanˆ kai o mhdenikìc). Gia sunart seic f i ston q ro L p i Rn i ) jewroôme to olokl rwma kai orðzoume I Z f 1,..., f k ) = Zx) R M k f i B i x)dx) i=1 C z := sup{i Z f 1,..., f k ) : f i pi = 1 gia i = 1,..., k}.

24 4 KEFŸALAIO. BASIKŸES ŸENNOIES Tìte C Z orðzetai an periorðsoume tic f i na eðnai sunart seic Gauss, dhl., C z := sup{i Z f 1,..., f k ) : f i pi = 1 kai f i x) = exp[ x, J i x)] me J i ènac pragmatikìc, summetrikìc, jetikˆ-orismènoc n i n i pðnakac }. An kai h bèltisth stajerˆ C Z den dðnetai akrib c, h parapˆnw sqèsh perièqei ènan algìrijmo gia ton upologismì twn C Z afoô ta oloklhr mata twn sunart sewn Gauss eðnai upologðsima. H apìdeixh thc genikeumènhc anisìthtac Young akìma kai qwrðc thn bèltisth stajerˆ eðnai arketˆ pio polôplokh apì thn apìdeixh thc sun jouc anisìthtac Young. H ˆllh basik oloklhrwtik anisìthta eðnai h bèltisth anisìthta Hardy-Littlewood-Sobolev [46], [60] kai [61]). Je rhma.1.. [61]) Anisìthta Hardy - Littlewood - Sobolev) 'Estw p, r > 1 kai 0 < λ < n me 1/p + λ/n + 1/r =. 'Estw f L p ) kai h L r ). Tìte upˆrqei mða bèltisth stajerˆ Cn, λ, p), anexˆrthth twn f kai h, ètsi ste fx) x y λ hy)dxdy Cn, λ, p) f p h r.4) H bèltisth stajerˆ ikanopoieð thn sqèsh Cn, λ, p) n n λ ω n 1/n) λ/n 1 pr ) λ/n λ/n + 1 1/p λ/n ) ) λ/n 1 1/r.5) ìpou ω n 1 = π n/ /Γ n ) eðnai o ìgkoc thc Sn 1 1).

25 .1. OLOKLHRWTIKŸES ANISŸOTHTES 5 An p = r = n/n λ), tìte [ ] 1+λ/n Cn, λ, p) = Cn, λ) = π λ/γn/ λ/) Γn/) Γn λ/) Γn).6) Se aut n thn perðptwsh upˆrqei isìthta sthn anisìthta.4) an kai mìnon an h const)f kai fx) = Aµ + x x 0 ) ) n λ)/.7) gia kˆpoia A C, 0 µ R kai x 0. Parathr seic : 1) H anisìthta.4) ìqi sthn bèltisth morf ) apodeðqjhke to 198 apì touc Hardy-Littlewood [46]) kai apì ton Sobolev [83]) to H bèltisth stajerˆ pou dðnetai apì thn.6) apodeðqjhke to 1983 apì ton E. Lieb [60]). ) H anisìthta.4) merikèc forèc anafèretai sthn bibliografða kai wc asjen c anisìthta Young. MporoÔme na parathr - soume ìti h.4) faðnetai san na eðnai h anisìthta Young me gx) na èqei antikatastajeð apì thn x λ. Aut h sunˆrthsh, wstìso, den eðnai se kˆje L p -q ro, allˆ par' ìla autˆ èqoume mða anisìthta anˆlogh thc anisìthtac Young. O ìroc asjen c kajðstatai apì to gegonìc ìti x λ eðnai ston asjen L q -q ro L q w ) me q = n/λ. Autìc o q roc orðzetai wc o q roc ìlwn twn metr - simwn sunart sewn f ètsi ste sup αmes{x : fx) > α}) 1/q <. α>0 Kˆje sunˆrthsh tou L q ) an kei ston q ro L q w ). Aplˆ mporoôme na parathr soume ìti f q q fx) q dx α q mes{x : fx) > α}. f >α H nìrma norm) tou q rou L q w ) dðnetai apì thn

26 6 KEFŸALAIO. BASIKŸES ŸENNOIES f q,w = supmesa) 1/r fx) dx, A A ìpou 1/q+1/r = 1 kai A eðnai èna tuqaðo metr simo sônolo mètrou mesa = A <. H sunˆrthsh x λ eðnai ston q ro L n/λ w ) kai [61]) x λ n/λ,w = n n λ [ω n 1/n] λ/n.8) Ed ω n 1 eðnai o ìgkoc thc monadiaðac sfaðrac S n 1, dhlad S n 1 = {x : x = 1}. H genik asjen c anisìthta Young dhl nei ìti gia g L q w ) kai 1 < p, q, r < + me 1/p + 1/q + 1/r =, h akìloujh anisìthta isqôei : fx)gx y)hy)dxdy C p,q,r f p g q,w h r. MporoÔme epðshc na sunˆgoume apì thn anisìthta Hardy - Littlewood -Sobolev ìti h sunèlixh eðnai mða fragmènh apeikìnish apì ton q ro L p ) L q w ) ston q ro L r ) ìpou L q w ) eðnai o asjen c L q - q roc. Me ˆlla lìgia, paðrnontac λ = n/q kai antikajist ntac to r me to r sthn.4), g h r 1 q me 1 p + 1 q = r. n ω n 1 ) 1/q Cn, n/q, p) g q,w h p.9) SÔmmorfoi metasqhmatismoð kai stereografik probol Kˆpoiec apì tic summetrðec thc.4) eðnai profaneðc. An antikatast soume tic fx) kai hx) me τ α f)x) := fx α) kai

27 .1. OLOKLHRWTIKŸES ANISŸOTHTES 7 τ α h)x) := hx α) gia α, mporoôme na diapist oume ìti ta dôo mèrh thc.4) den allˆzoun thn tim touc. OmoÐwc gia R On), ìpou On) eðnai h orjog nia omˆda twn strèyewn kai anaklˆsewn tou, mporoôme na antikatast soume tic f kai h me Rf)x) := fr 1 x) kai Rh)x) := hr 1 x) kai pˆli na parathr soume ìti oi timèc twn mer n thc.4) den allˆzoun. 'Etsi h anisìthta Hardy - Littlewood - Sobolev eðnai analloðwth kˆtw apì th drˆsh thc EukleÐdeiac omˆdac [R, α), fx)] fr 1 x α), R On), α kai omoðwc gia thn h. 'Allh mða apl summetrða eðnai h bajmwt summetrða. An antikatast soume fx), hx) me s p/n fsx), s q/n hsx) gia s > 0, tìte h.4) eðnai pˆli analloðwth. H sômmorfh omˆda eðnai h omˆda twn metasqhmatism n h opoða diathreð tic gwnðec. Upˆrqoun pollèc apeikonðseic oi opoðec diathroôn tic gwnðec kai mða apì autèc eðnai h antistrof thc monadiaðac sfaðrac, I :, x x =: Ix)..10) x Upˆrqoun pollèc parathr seic pou mporoôme na kˆnoume gia thn antðstrofh apeikìnish. Pr ta ap' ìla ìti aut den eðnai orismènh se ìlo to allˆ sto qwrðc thn arq. Wstìso mporoôme na epekteðnoume thn apeikìnish I sto Ṙn, pou eðnai h enìc shmeðou sumpagopoðhsh tou dhlad Ṙn = { }, ìpou orðzetai na eðnai to stoiqeðo to opoðo perièqetai se ìla ta mh fragmèna anoiktˆ sônola. An orðsoume I0) = kai I ) = 0, I epekteðnetai sto Ṙn. T ra shmei noume ìti Ix) Iy) = x x y y = 1 x xy x y + 1 y

28 8 KEFŸALAIO. BASIKŸES ŸENNOIES = 1 x 1 y x y..11) An epilèxoume dôo C 1 kampôlec xt), yt) ston Ṙn me x0) = y0) = z 0, tìte ut) := Ixt)) kai υt) := Iyt)) orðzoun dôo nèec kampôlec ston Ṙn. 'Eqoume na elègxoume ìti h gwnða metaxô twn efaptìmenwn dianusmˆtwn twn ut) kai υt) ta opoða shmei noume me u kai υ) èqei thn Ðdia tim sto t = 0 me thn gwnða metaxô twn ẋt) kai ẏt) sto t = 0. Allˆ apì thn.11) èqoume 1 u υ = lim t 0 t Ixt) Iz)+Iz) Iyt)) = 1 z ẋ ẏ.1) kai, eidikìtera, u = ẋ / z, υ = ẏ / z, apì thn opoða brðskoume ìti u υ u υ = ẋẏ ẋ y dhlad, I eðnai sômmorfh. Upˆrqei mða polô ìmorfh perigraf tou Ṙn me thn ènnoia thc stereografik c probol c. OrÐzoume s = s 1, s,..., s n+1 ) me s i = x i 1 + x για i = 1,..., n και s n+1 = 1 x 1 + x..13) An x =, tìte s i = 0 gia i = 1,..., n kai s n+1 = 1. 'Enac aplìc upologismìc deðqnei ìti n+1 i=1 s i = 1. 'Etsi S : x s eðnai mða apeikìnish apì to Ṙn S n. H antðstrofh thc S dðnetai apì thn x i = s i 1 + s n+1 i = 1,..., n,.14) kai ja kaloôme tic suntetagmènec thc S 1 suntetagmènec gia thn S n. stereografikèc

29 .1. OLOKLHRWTIKŸES ANISŸOTHTES 9 'Enac aplìc upologismìc deðqnei ìti n+1 s i t i ) = s t = i= x )1 + y ) x y,.15) ìpou s = Sx) kai t = Sy) kai S eðnai sômmorfh apeikìnish. MÐa isometrða enìc q rou, mil ntac genikˆ, eðnai mða apeikìnish h opoða diathreð tic apostˆseic metaxô twn shmeðwn, dhlad mða isometrða tou L p Ω) eðnai mða apeikìnish twn sunart sewn h opoða diathreð thn nìrma norm) f g p. To sônolo twn isometri n thc sfaðrac S n eðnai h omˆda On + 1). An γ : Ṙn Ṙn eðnai sthn sômmorfh omˆda C, tìte mporoôme na orðsoume th drˆsh thc γ pˆnw se sunart seic f ston L p ) wc akoloôjwc. Epilègoume mða akoloujða f k L p ) ètsi ste f k mhdenðzetai èxw apì thn mpˆlla B k gia ìla ta k = 1,,... kai ètsi ste f k f ston L p ). ParathroÔme ìti γ f k )x) := J γ 1x) 1/p f k γ 1 x).16) eðnai kalˆ orismènh gia ìla ta k. Ed J γ 1x) eðnai h Iakwbian tou metasqhmatismoô γ 1. Aut h apeikìnish γ eðnai grammik kai apì thn allag twn metablht n èqoume γ f k p = f k p..17) 'Etsi akoloujeð ìti γ f k sugklðnei isqurˆ ston L p ) se mia sunˆrthsh γ f kai autì to ìrio eðnai anexˆrthto thc proseggistik c akoloujðac f k. Me ton Ðdio trìpo mporoôme na shk soume sunart seic tou L p ) sthn sfaðra S n. Aplˆ orðzoume Pˆli F s) = S f)s) = J S 1s) 1/p fs 1 s))..18)

30 30 KEFŸALAIO. BASIKŸES ŸENNOIES F Lp S n ) = f Lp )..19) EÐnai aparaðthto na upologðsoume thn Iakwbian thc stereografik c probol c J S x). H metrik pˆnw sthn sfaðra S n ekfrˆzetai se ìrouc twn stereografik n suntetagmènwn apì thn sqèsh ) g ij = δ ij..0) 1 + x Wc ek toôtou to stoiqeðo ìgkou pˆnw sthn sfaðra S n eðnai kai sunep c ds = ) n dx.1) 1 + x J S x) = ) n και J 1 + x S 1s) = 1 + s n+1) n..) Je rhma.1.3. [61]) SÔmmorfh analloi thta thc anisìthtac Hardy-Littlewood-Sobolev) Upojètoume ìti p = r sthn.4) kai ìti F L p S n ) kai f L p ) sqetðzontai me thn.18). 'Estw H kai h na eðnai èna ˆllo zeugˆri susqetismèno me ton Ðdio trìpo. Tìte kai fx) x y λ hy)dxdy = S n S n F s) s t λ Ht)dsdt.3) F p = f p..4) Ed s t = n+1 i=1 s i t i ) eðnai h EukleÐdeia apìstash tou +1 kai ìqi h gewdaisiak apìstash thc S n ). Autì deðqnei thn

31 .1. OLOKLHRWTIKŸES ANISŸOTHTES 31 analloi thta kˆtw apì ìlec tic summetrðec thc S n, dhlad thn analloi thta kˆtw apì thn omˆda On+1). Epiplèon, h anisìthta Hardy-Littlewood-Sobolev eðnai sômmorfa analloðwth, dhlad, gia γ C γ f)x) x y λ γ h)y)dxdy R n = fx) x y λ hy)dxdy, kai γ f p = f p, γ h p = h p.

32 3 KEFŸALAIO. BASIKŸES ŸENNOIES. O Metasqhmatismìc Fourier O Metasqhmatismìc Fourier eðnai èna shmantikì ergaleðo sthn Anˆlush, polô agaphtì apì touc analôstec, touc fusikoôc epist monec kai touc mhqanikoôc. H axða tou MetasqhmatismoÔ Fourier eðnai ìti allˆzei touc telestèc thc diafìrishc se pollaplasiastikoôc telestèc. Orismìc..1. Orismìc tou L 1 metasqhmatismoô Fourier 'Estw f na eðnai mða sunˆrthsh ston L 1 ). O metasqhmatismìc Fourier thc f, pou shmei netai me f, eðnai h sunˆrthsh ston pou dðnetai apì thn sqèsh fk) = e πik,x) fx)dx ìpou k, x) := n k i x i. Oi akìloujec algebrikèc idiìthtec eðnai ta basikˆ ergaleða sth melèth tou metasqhmatismoô Fourier : H apeikìnish f f eðnai grammik τ h fk) = e πk,h) fk), h R n δ λ fk) = λ n fλk), λ > 0 ìpou τ h eðnai o metajetikìc telest c τ h f)x) = fx h), kai δ λ eðnai o bajmwtìc telest c, δ λ f)x) = fx/λ). EÐnai eôkolo na apodeðxoume ìti f L ) kai f f 1 kai epðshc ìti f eðnai mða suneq c kai wc ek toôtou metr simh) sunˆrthsh. An f kai g eðnai sunart seic ston L 1 ) tìte apì to Je rhma Fubini Je rhma 1.1 sto biblðo [61]) f g L 1 ) kai epðshc apì to Ðdio Je rhma, i=1

33 .. O METASQHMATISMŸOS FOURIER 33 f g)k) = e πik,x) fx y)gy)dydx = e πik,y) gy) e πik,x y)) fx y)dxdy = fk)ĝk)..5) Je rhma... [61]) Metasqhmatismìc Fourier thc sunˆrthshc tou Gauss) Gia λ > 0, shmei noume me g λ th sunˆrthsh tou Gauss pˆnw sto pou eðnai gia x. Tìte g λ x) = e πλ x ĝ λ k) = λ n/ e π k /λ Je rhma..3. [61]) Je rhma tou Plancherel) An f L 1 ) L ), tìte f eðnai ston L ) kai o akìloujoc tôpoc tou Plancherel isqôei f = f. H apeikìnish f f èqei mða monadik epèktash se mða suneq grammik apeikìnish apì ton L ) ston L ) h opoða eðnai mða isometrða, dhlad o parapˆnw tôpoc tou Plancherel isqôei gia aut n thn epèktash. SuneqÐzoume na shmei noume aut n thn apeikìnish f f akìma kai an f / L 1 )). An f kai g eðnai ston L ), tìte isqôei o tôpoc tou Parseval f, g) := fx)gx)dx = fk)ĝk)dk = f, ĝ).

34 34 KEFŸALAIO. BASIKŸES ŸENNOIES Orismìc..4. Orismìc tou L metasqhmatismoô Fourier 'Estw f L ). O q roc L 1 ) L ) eðnai puknìc ston L ) dhlad upˆrqei mða akoloujða f j L 1 ) L ) tètoia ste f j f 0 kaj c j. Apì to tôpo tou Plancherel èqoume ìti f j L ) kai f j f m = f j f m 0 kaj c j, m. Epomènwc h akoloujða f j eðnai akoloujða Cauchy ston L ) kai epeid o q roc L ) eðnai pl rhc ja sugklðnei se kˆpoia sunˆrthsh ston L ) thn opoða sumbolðzoume me f. Aut loipìn h sunˆrthsh f lègetai metasqhmatismìc Fourier thc f ston L ). Je rhma..5. [61]) TÔpoc thc antistrof c) Gia f L ), orðzoume Tìte f x) := f x) f = f). Je rhma..6. [61]) H bèltisth anisìthta Hausdorff- Young) 'Estw 1 < p < kai èstw f L p ) L 1 ). Tìte, me 1/p + 1/p = 1, ìpou f p C n p f p C p = [p 1/p p ) 1/p ] eðnai h bèltisth stajerˆ. Epiplèon, isìthta sthn parapˆnw anisìthta epitugqˆnetai an kai mìno an f eðnai h sunˆrthsh tou Gauss thc morf c fx) = Ae x,mx)+b,x)

35 .. O METASQHMATISMŸOS FOURIER 35 me A C, M kˆje summetrikìc, pragmatikìc, jetikˆ orismènoc pðnakac kai B kˆje diˆnusma tou C n. Orismìc..7. Orismìc tou L p metasqhmatismoô Fourier Qrhsimopoi ntac thn anisìthta Hausdorff-Young mporoôme na orðsoume ton metasqhmatismì Fourier ston L p ) gia 1 < p <. 'Estw f L p ). O q roc L 1 ) L p ) eðnai puknìc ston L p ) dhlad upˆrqei mða akoloujða f j L 1 ) L p ) tètoia ste f j f p 0 kaj c j. Apì thn anisìthta Hausdorff-Young èqoume ìti f j L ) kai f j f m p C n p f j f m p 0 kaj c j, m. Epomènwc h akoloujða f j eðnai akoloujða Cauchy ston L p ) kai epeid o q roc L p ) eðnai pl rhc ja sugklðnei se kˆpoia sunˆrthsh ston L p ) thn opoða sumbolðzoume me f. Aut loipìn h sunˆrthsh f lègetai metasqhmatismìc Fourier thc f ston L p ). Je rhma..8. [61]) Metasqhmatismìc Fourier gia SunelÐxeic) 'Estw f L p ) kai g L q ), kai èstw 1 + 1/r = 1/p + 1/q. Upojètoume 1 p, q, r. Tìte f g)k) = fk)ĝk). Je rhma..9. [61]) Metasqhmatismìc Fourier thc x α n ) 'Estw f na eðnai mða sunˆrthsh ston Cc ) kai èstw 0 < α < n. Tìte an α ) γ α := π α/ Γ, γ α k α fk) ) x) = γn α x y α n fy)dy Je rhma..10. [61]) Metasqhmatismìc Fourier thc x α n f gia f L p ))

36 36 KEFŸALAIO. BASIKŸES ŸENNOIES An 0 < α < n kai an f L p ) me p = n n+α, tìte apì thn anisìthta Hausdorff-Young f upˆrqei. Epiplèon, h sunˆrthsh g := γ n α x α n f eðnai mða L )-sunˆrthsh apì thn anisìthta Hardy-Littlewood- Sobolev Je rhma.1.) kai wc ek toôtou èqei èna metasqhmatismì Fourier ĝ. To nèo apotèlesma eðnai h sqèsh metaxô ĝ kai f Epiplèon, γ α k α fk) = ĝk). γ α k α fk) dk = γ n α fx)fy) x y α n dxdy

37 .3. QŸWROI SOBOLEV 37.3 Q roi Sobolev Test Sunart seic O Q roc DΩ)) 'Estw Ω na eðnai èna anoiktì sônolo sto eidikìtera Ω mporeð na eðnai to Ðdio to ). UpenjumÐzoume ìti C c Ω) eðnai o q roc ìlwn twn apeðrwc diaforðsimwn sunart sewn, twn opoðwn to st rigma eðnai sumpagèc uposônolo tou Ω. UpenjumÐzoume epðshc ìti to st rigma support) miac suneqoôc sunˆrthshc orðzetai na eðnai to perðblhma tou sunìlou pˆnw sto opoðo h sunˆrthsh den mhdenðzetai. O q roc twn test sunart sewn, DΩ), perièqei ìlec tic sunart seic tou q rou Cc Ω) me thn epiprìsjeth ènnoia thc sôgklishc : MÐa akoloujða φ m C c Ω) sugklðnei ston DΩ) sthn sunˆrthsh φ C c Ω) an kai mìnon an upˆrqei kˆpoio sumpagèc sônolo K Ω ètsi ste to st rigma thc φ m φ eðnai mèsa sto K gia ìla ta m kai, gia kˆje epilog twn mh arnhtik n akeraðwn α 1,..., α n, ) α1 ) αn φ m ) α1 ) αn φ x 1 x n x 1 x n omoiìmorfa sto K kaj c m. Lème ìti mða akoloujða sunart sewn ψ m sugklðnei sthn ψ omoiìmorfa sto K ìtan sup ψ m x) ψx) 0 x K kaj c m. DΩ) eðnai grammikìc q roc, dhlad sunart seic mporoôn na prostðjentai kai na pollaplasiˆzontai me stajerèc. Orismìc twn katanom n MÐa katanom distribution) T eðnai èna suneqèc grammikì suna-

38 38 KEFŸALAIO. BASIKŸES ŸENNOIES rthsiakì sto DΩ), dhlad T : DΩ) C ètsi ste gia φ, φ 1, φ DΩ) kai λ C T φ 1 + φ ) = T φ 1 ) + T φ ) και T λφ) = λt φ) kai suneqèc shmaðnei ìti an φ n DΩ) kai φ n φ DΩ) tìte T φ n ) T φ). Oi katanomèc mporeð na prostðjentai kai na pollaplasiˆzontai me stajerèc. Autìc o grammikìc q roc shmei netai me D Ω) kai onomˆzetai o duðkìc tou q rou DΩ). H ènnoia thc sôgklishc twn katanom n eðnai shmeiak : MÐa akoloujða katanom n T j DΩ) sugklðnei ston D Ω) an, gia kˆje φ DΩ), oi arijmoð T j φ) sugklðnoun ston T φ). Orismìc tou q rou L p loc Ω) O q roc twn topik c p-oloklhr simwn sunart sewn, L p loc Ω) gia 1 p, apoteleðtai apì ekeðnec tic Borel metr simec sunart seic pˆnw sto Ω me thn idiìthta ìti f Lp K) < gia kˆje sumpagèc sônolo K Ω. IsodÔnama, arkeð na isqôei f Lp K) < ìtan K eðnai kˆje kleist mpˆlla tou Ω. MÐa akoloujða sunart sewn f 1, f,... tou q rou L p loc Ω) lègetai ìti sugklðnei isqurˆ sthn f L p loc Ω) an f j f ston q ro L p K) me thn sun jh ènnoia pou anafèrjhke parapˆnw gia kˆje sumpagèc sônolo K Ω. ParomoÐwc, f j sugklðnei asjen c sthn f an f j f asjen c se kˆje L p K). O q roc L p loc Ω) eðnai ènac dianusmatikìc q roc allˆ den èqei mða aplˆ orismènh nìrma. Epiplèon, f L p loc Ω) den sunepˆgetai ìti f L p Ω). Profan c, L p loc Ω) Lp Ω) kai an r > p, èqoume apì thn anisìthta Hölder

39 .3. QŸWROI SOBOLEV 39 L p loc Ω) Lr locω), kai eðnai lˆjoc na jewroôme ìti L p Ω) L r Ω), ektìc kai an Ω èqei peperasmèno mètro. 'Oso aforˆ tic katanomèc, L 1 loc Ω) eðnai o pio shmantikìc q roc. 'Estw f na eðnai mða sunˆrthsh ston q ro L 1 Ω). loc Gia kˆje φ DΩ) èqei ènnoia na jewr soume T f φ) := Ω fφdx,.6) to opoðo profan c orðzei èna grammikì sunarthsiakì ston q ro DΩ). To sunarthsiakì T f eðnai epðshc suneqèc afoô T f φ) T f φ m ) = Ω φx) φ m x))fx) fx) dx, sup φx) φ m x) x K to opoðo teðnei sto 0 exaitðac thc omoiìmorfhc sôgklishc twn φ m. 'Etsi T f eðnai ston D Ω). An mða katanom T dðnetai apì thn.6) gia kˆpoia f L 1 loc Ω), lème ìti h katanom T eðnai h sunˆrthsh f. 'Ena axioshmeðwto parˆdeigma miac katanom c, h opoða den eðnai aut c thc morf c eðnai h legìmenh dèlta-sunˆrthsh tou Dirac δ x φ) = φx) me x Ω stajerì. EÐnai profanèc ìti δ x D Ω). 'Etsi, to d-mètro tou Dirac { 1 αν x A δ x A) = 0 αν x / A ìpwc kˆje Borel mètro, mporeð na jewrhjeð san mða katanom. K

40 40 KEFŸALAIO. BASIKŸES ŸENNOIES An kai o q roc twn test sunart sewn DΩ), eðnai mða polô periorismènh klˆsh sunart sewn, o duðkìc tou q roc D Ω) eðnai arketˆ megˆloc, ìpwc faðnetai apì to akìloujo je rhma Je rhma.3.1. [61]) 'Estw Ω eðnai èna anoiktì sônolo kai èstw f kai g na eðnai sunart seic ston L 1 loc Ω). Upojètoume ìti oi katanomèc pou orðzontai apì tic f kai g eðnai Ðsec, dhlad fφ = gφ Ω gia ìlec tic φ DΩ). Tìte fx) = φx) gia sqedìn kˆje x Ω. Parˆgwgoi twn katanom n DÐnoume t ra thn ènnoia thc wc proc tic katanomèc parag gou asjenoôc parag gou). 'Estw T D Ω) kai èstw α 1,..., α n na eðnai mh arnhtikoð akèraioi. OrÐzoume thn katanom ) α1 ) α ) αn x 1 x x n T, thn opoða shmei noume me D α T, me thn drˆsh thc pˆnw se kˆje φ DΩ) wc akoloôjwc: Ω me D α T )φ) = 1) α T D α φ) α = i α i. To sômbolo i T sumbolðzei to D α sthn eidik perðptwsh α i = 1, α j = 0 gia j i. Me T, shmei noume thn katˆ katanom klðsh distributional gradient) thc T pou eðnai h n-ada 1 T, T,..., n T ). An f eðnai mða C α Ω)-sunˆrthsh ìqi aparaðthta sumpagoôc sthrðgmatoc), tìte D α T f )φ) := 1) α Ω D α φ)fdx = D α f)φdx =: T Dα fφ), Ω

41 .3. QŸWROI SOBOLEV 41 ìpou h mèsh isìthta isqôei apì thn katˆ parˆgontec olokl rwsh. Ja deðxoume ìti D α T eðnai prˆgmati mða katanom. Profan c eðnai grammik, ètsi mènei na exetasteð h sunèqeia sto DΩ). 'Estw φ m φ ston DΩ). Tìte D α φ m D α φ sto DΩ) afoô kai supp{d α φ m D α φ} supp{φ m φ} K D β D α φ m D α φ) = D β+α φ m D β+α φ sugklðnei sto 0 omoiìmorfa pˆnw sta sumpag sônola ed β + α aplˆ sumbolðzei ton poludeðkth β 1 +α 1, β +α,...β n +α n ). 'Etsi, D α φ kai D α φ m eðnai sunart seic ston DΩ) me D α φ m D α φ ìtan m. Wc ek toôtou, kaj c m, D α T )φ m ) := 1) α T D α φ m ) 1) α T D α φ) =: D α T )φ) To sômbolo l T sumbolðzei to sônolo ìlwn twn katanom n {D α T }, gia ìla ta α me α = l. AxÐzei na shmei soume ìti h ènnoia thc asjenoôc parag gou epekteðnei thn klassik ènnoia thc parag gou kai sumfwneð me aut ìtan h klassik parˆgwgoc upˆrqei kai eðnai suneq c. Telei noume aut thn parˆgrafo deðqnontac ìti h diafìrish twn katanom n eðnai suneq c telest c ston q ro D Ω). Prˆgmati, an T j φ) T φ) gia ìlec tic φ DΩ), tìte, apì ton orismì thc parag gou miac katanom c D α T j )φ) = 1) α T j D α φ) 1) α T D α φ) = D α T )φ) ìtan j afoô D α φ DΩ). Orismìc.3.. O q roc Sobolev W l,p gia l > 0 akèraio. 'Estw Ω na eðnai mða fragmènh mh fragmènh perioq tou. Gia 1 < p < + kai akèraio l > 0, orðzoume W l,p Ω) na eðnai o dianusmatikìc q roc twn stoiqeðwn tou L p Ω) twn opoðwn

42 4 KEFŸALAIO. BASIKŸES ŸENNOIES oi merikèc parˆgwgoi me thn ènnoia twn katanom n) mèqri l tˆxhc an koun ston q ro L p Ω). 'Etsi, W l,p Ω) eðnai ènac Banach q roc, o opoðoc eðnai efodiasmènoc me thn nìrma norm) f W l,p Ω) = D α f p L p Ω) α l Eidikìtera, W 0,l Ω) = L p Ω) kai W l,p Ω) eðnai ènac upìqwroc tou W l 1,p. 'Estw t ra, γ na eðnai ènac pragmatikìc arijmìc pou ikanopoieð 0 < γ < 1. Shmei noume me W γ,p Ω) to dianusmatikì q ro twn stoiqeðwn tou L p Ω) gia ta opoða h hminìrma seminorm) eðnai peperasmènh. f W γ,p Ω) = Ω Ω 1/p fx) fy) p ) 1/p dxdy x y n+γp Orismìc.3.3. O q roc Sobolev W l+γ,p gia l > 0 akèraio kai 0 < γ < 1. 'Estw Ω na eðnai mða fragmènh mh fragmènh perioq tou. Gia 1 p < +, akèraio l > 0 kai 0 < γ < 1, orðzoume W l+γ,p Ω) na eðnai o dianusmatikìc q roc twn stoiqeðwn tou L p Ω) twn opoðwn oi merikèc parˆgwgoi tˆxhc l an koun ston q ro W γ,p Ω). 'Etsi, W l+γ,p Ω) eðnai ènac Banach q roc, o opoðoc eðnai efodiasmènoc me thn nìrma norm) f W l+γ,p Ω) = f p W l,p Ω) + D α f p W γ,p Ω) α =l Epeid l = {D α } ìpou α = l h parapˆnw sqèsh grˆfetai f W l+γ,p Ω) = f p W l,p Ω) + lf p W γ,p Ω)) 1/p.. 1/p.

43 .3. QŸWROI SOBOLEV 43 'Etsi orðzoume ton q ro Sobolev W α,p Ω) gia kˆje pragmatikì arijmì α > 0. An [α] kai {α} eðnai antðstoiqa to akèraio kai dekadikì mèroc tou α, tìte f W α,p Ω) = f pw [α],pω) + Ω Ω [α]fx) [α]fy) p ) 1/p dxdy. x y n+{α}p MporoÔme na parathr soume ìti o q roc twn apeðrwc diaforðsimwn sunart sewn me sumpagèc st rigma Cc Ω) perièqetai se kˆje ènan apì touc q rouc W α,p Ω). 'Estw W α,p 0 Ω) na eðnai h pl rwsh tou q rou Cc Ω) ston q ro W α,p Ω). Tìte W α,p 0 Ω) eðnai ènac kleistìc grammikìc upìqwroc tou W α,p Ω). 'Eqei apodeiqjeð ìti W α,p 0 Ω) = W α,p Ω) an 0 α 1/p. Akìma eðnai W α,p 0 ) = W α,p ) gia kˆje α kai p. Orismìc.3.4. O suzug c tou q rou Sobolev W α,p Upojètoume ìti 1 < p < + kai èstw p na eðnai o suzug c tou p, dhlad p = p/p 1) 1/p+1/p = 1 ètsi 1 < p < + ). Gia α > 0, orðzoume W α,p Ω) na eðnai o suzug c q roc dual space) tou W α,p 0 Ω), dhlad o q roc twn suneq n grammik n morf n pˆnw ston W α,p 0 Ω). 'Etsi W α,p Ω) eðnai ènac q roc Banach. Eidikˆ gia thn perðptwsh p =, eðnai perissìtero dhmofilèc na qrhsimopoioôme ton sumbolismì H α Ω) antð tou sumbolismoô W α, Ω) kai epðshc axðzei na shmei soume ìti H α Ω) eðnai q roc Hilbert. O deðkthc α tou q rou Sobolev W α,p Ω) W α,p 0 Ω)) kaleðtai tˆxh order).

44 44 KEFŸALAIO. BASIKŸES ŸENNOIES

45 Kefˆlaio 3 Oi q roi H s ) kai H s,p ) 3.1 O q roc H s ) Se ìti akoloujeð sumbolðzei ton telest tou Laplace ston. Orismìc [84]) 'Estw s ènac jetikìc pragmatikìc arijmìc. Oi telestèc ) s/ orðzontai se q rouc Fourier dhlad se q rouc me sunart seic pou èqoun metasqhmatismì Fourier wc ex c ) s/ f) k) := π k ) s fk). 'Otan 0 < s < n kai f L p ), p 1, h exðswsh ) s/ φ = f èqei mða monadik lôsh ston q ro L q ) me 1 q = 1 p s n. Aut h lôsh dðnetai apì to dunamikì tou Riesz Riesz Potential) [84]) φ = ) s/ f = 1 c s x y s n fy)dy me c s = s π n/ Γ ) s Γ n s ). O pur nac tou Riesz Riesz kernel) ston eðnai : 45

46 46 KEFŸALAIO 3. OI QŸWROI H S R N ) KAI H S,P R N ) I s x) = x s n c s, 0 < s < n. H sunˆrthsh tou Green gia ) s/ eðnai I s x y), gia 0 < s < n. Orismìc 'Estw s ènac jetikìc pragmatikìc arijmìc. MÐa sunˆrthsh f C c ) lègetai ìti an kei ston q ro H s ) an kai mìno an f H s ) : = )s/ f + f = fk) 1 + π k ) s )dk < O q roc H s ) eðnai efodiasmènoc me to eswterikì ginìmeno f, g) Hs ) = fk)ĝk)1 + π k ) s )dk. An f kai g eðnai ston H s ) tìte h morf sesquilinear form) f, ) s g) = π k ) s fk)ĝk)dk èqei ènnoia apì thn anisìthta Hölder. EpÐshc f, ) s f) = ) s/ f. Orismìc [84]) 'Estw s ènac jetikìc pragmatikìc arijmìc. Oi telestèc I ) s/ orðzontai se q rouc Fourier dhlad se q rouc me sunart seic pou èqoun metasqhmatismì Fourier wc ex c I ) s/ f) k) := 1 + π k ) ) s/ fk) EÐnai eôkolo na apodeiqjeð ìti upˆrqoun stajerèc α, A ètsi ste

47 3.1. O QŸWROS H S R N ) 47 α I ) s/ f ) s/ + f A I ) s/ f EpÐshc f, I ) s f) = I ) s/ f. 'Otan 0 < s < n h exðswsh I ) s/ φ = f èqei mða monadik lôsh ston q ro L q ). Aut h lôsh dðnetai apì to dunamikì tou Bessel Bessel Potential) [44]) ìpou φ = I ) s/ f = G s f 1 1 G s = e x 4π) n/ 4δ e δ dδ δ n+s Γs/) 0 δ eðnai o pur nac tou Bessel Bessel kernel) ston. H sunˆrthsh tou Green gia I ) s/ eðnai G s x ) y). An lˆboume upìyh to gegonìc ìti e δ 4π = 1 + o e δ 4π, δ 0, èqoume ìti G s x) = x n+s c s + o x n+s ), kaj c x 0 an 0 < s < n. EpÐshc, G s x) = Oe c x ) kaj c x, gia kˆpoio c > 0, dhlad o pur nac G s fjðnei kaj c x. O pur nac G s mporeð epðshc na dojeð wc mða trðtou eðdouc sunˆrthsh Bessel bl. [65]). Sto shmeðo autì upenjumðzoume thn sunˆrthsh tou Green gia ton telest + µ me µ > 0. GnwrÐzoume kalˆ bl. [61]) ìti gia kˆje n 1 kai µ > 0 upˆrqei mða sunˆrthsh G µ y h opoða ikanopoieð thn exðswsh kai h opoða dðnetai apì thn sqèsh + µ )G µ y = δ y. 3.1)

48 48 KEFŸALAIO 3. OI QŸWROI H S R N ) KAI H S,P R N ) G µ yx) = G µ x y), 3.) G µ x) = 0 4πt) n/ e x 4t µt dt. 3.3) H sunˆrthsh G µ y kaleðtai dunamikì Yukawa Yukawa potential), toulˆqiston ìtan n = 3, kai èpaixe shmantikì rìlo sthn jewrða twn stoiqeiwd n swmatidðwn mesons), gia thn opoða o H. Yukawa kèrdise to brabeðo Nobel. H sunˆrthsh G µ èqei tic akìloujec idiìthtec [61]) i) G µ > 0 gia ìla ta x. ii) R G µ x)dx = µ. n iii) An f L p ), gia kˆpoio 1 p, tìte ux) = G µ f = G µ yx)fy)dy 3.4) eðnai ston q ro L r ) kai ikanopoieð thn exðswsh + µ )u = f 3.5) me thn ènnoia twn katanom n kai gia katˆllhla p, r se kˆje perðptwsh. Sugkekrimèna: Gia n = 1, p r. Gia n =, p r ìtan p > 1 kai 1 r < ìtan p = 1. Gia n = 3, p r np/n p) ìtan 1 < p < n/, p r ìtan p n/ kai 1 r < n/n ) ìtan p = 1. Epiplèon, 3.4) eðnai h monadik lôsh thc 3.5) me thn idiìthta ìti aut eðnai ston q ro L r ) gia kˆpoio r 1. iv) O metasqhmatismìc Fourier thc G µ eðnai Ĝ µ k) = π k ) + µ ) )

49 3.. IDIŸOTHTES TOU QŸWROU H S R N ) Idiìthtec tou q rou H s ) Prìtash [85]) An f eðnai ston H l ), l jetikìc akèraioc, tìte D α fk) = πik) α fk) me α l. O q roc Sobolev W l, ) eðnai efodiasmènoc me thn nìrma norm) f W l, ) = D α f L ) 0 α l gia f : C me l ènac jetikìc akèraioc. Aut h nìrma eðnai isodônamh me thn nìrma Prˆgmati, f W l, ) = Allˆ, f H l ) = fk) 1 + π k ) l )dk. 0 α l = 0 α l = 0 α l D α f L ) = 0 α l D α f), D α f) ) L ) D α f, D α f) L ) πik) α fk)πik) α fk)dk = ) πik) α = πik 1 ) α 1 πikn ) α n = πik 1 ) α1 πik n ) α n = 1) α πik) α 'Etsi,

50 50 KEFŸALAIO 3. OI QŸWROI H S R N ) KAI H S,P R N ) ) = 0 α l = 0 α l = 0 α l = 0 α l = 0 α l = 0 α l 1) α πik) α πik) α fk) 1) α πik) α fk) 1) α πik 1 ) α1 πik n ) α n fk) 1) α i) α πk 1 ) α1 πk n ) α n fk) 1) α 1) α πk 1 ) α1 πk n ) α n fk) 4π k1) α1 4π k n ) α n fk) C π k ) l ) fk) = C 1 f H l ) gia C 1 arkoôntwc megˆlo. Apì thn ˆllh pleurˆ, an epilèxoume C arkoôntwc megˆlo tìte C f W l, ) f H l ) kai ètsi h isodunamða twn norm n apodeðqjhke. EpÐshc akìma genikìtera èqoume tic akìloujec dôo Protˆseic [65] sel. 347). Prìtash 3... [65]) 'Estw l = 1,,..., tìte upˆrqoun jetikoð arijmoð c kai C exart menoi apì ta n, p, l ètsi ste c ) l/ f Lp ) l f Lp ) C ) l/ f Lp ) gia ìlec tic f C c ), ìpou upenjumðzoume ìti

51 3.. IDIŸOTHTES TOU QŸWROU H S R N ) 51 l f Lp ) = D α fx) α =l p/ dx Prìtash [65]) 'Estw l = 1,,..., tìte upˆrqoun jetikoð arijmoð c kai C exart menoi apì ta n, p, l ètsi ste 1/p. c f W l,p ) I ) l/ f Lp ) C f W l,p ). Parat rhsh Sunduˆzontac thn Prìtash 3..1, thn Prìtash 3..3 kai thn Prìtash 3..4 èqoume ìti gia kˆje pragmatikì arijmì s > 0 W s, ) = H s ), ìpou W s, ) o q roc Sobolev, ton orismì tou opoðou d same sto Kefˆlaio, dhlad c α [s] C α [s] D α f + D α f + Rn [s] fx) [s] fy) dxdy x y n+{s} f H s ) Rn [s] fx) [s] fy) dxdy. x y n+{s} 'Allh mða shmantik Prìtash pou ja qrhsimopoi soume sthn sunèqeia eðnai h ex c: Prìtash [67], [8]) Gia 0 < γ < 1, èqoume ìti Rn fx) fy) dxdy = Cγ, n) π k ) γ ) R x y fk) dk n+γ n ìpou Cγ, n) eðnai mða stajerˆ.

52 5 KEFŸALAIO 3. OI QŸWROI H S R N ) KAI H S,P R N ) Prìtash [8]) 'Eqoume ìti H s 1 Rn ) H s Rn ) gia 0 < s < s 1. Apìdeixh f H s 1 ) = fk) 1 + π k ) s 1 )dk = fk) 1 + π k ) s 1 )dk k 1 + fk) 1 + π k ) s 1 )dk k >1 fk) 1 + π) s 1 )dk k 1 + fk) 1 + π k ) s 1 )dk k >1 1 + π) s 1 ) fk) dk + fk) 1 + π k ) s 1 )dk = ) k >1 An k > 1 tìte π k ) s 1 π k )s 1 + π k )s 1 ) 1 + π k ) s ). 'Etsi ) 1 + π) s ) f + fk) 1 + π k ) s )dk k >1 1 + π) s ) f + fk) 1 + π k ) s )dk = 1 + π) s ) f + f H s ) C f H s ) ìpou C eðnai mða stajerˆ. To epìmeno je rhma mac exasfalðzei thn puknìthta tou q rou C c ) stouc q rouc H s ) kai èqei anakoinwjeð sthn ergasða [8].

53 3.. IDIŸOTHTES TOU QŸWROU H S R N ) 53 Je rhma [8]) An f eðnai mða sunˆrthsh ston q ro H s ) tìte upˆrqei mða akoloujða sunart sewn pou an kei ston q ro C c ) ètsi ste f m f Hs ) 0 kaj c m. Apìdeixh Arqikˆ ja apodeðxoume to Je rhma gia s = l akèraio kai katìpin gia kˆje tuqìnta pragmatikì s > 0. 'Estw j : R + na eðnai ston q ro C c ) me j = 1 kai èstw j ε x) := ε n jx/ε) gia ε > 0. Tìte, afoô f kai D α f eðnai L ) - sunart seic gia α l diìti f eðnai ston q ro H l )), apì to Je rhma.16 tou biblðou [61] èqoume ìti f ε := j ε f f kai g ε := j ε D α f D α f sugklðnei isqurˆ ston L ) ìtan ε 0 gia α l. 'Etsi èqoume ìti f ε f isqurˆ ston H l ) diìti g ε = D α f ε. Allˆ autì kai pˆli eðnai alhjèc apì to Je rhma.16 tou biblðou [61]. EpÐshc apì to Ðdio je rhma h sunˆrthsh f ε eðnai mða sunˆrthsh tou q rou C ). Jètontac ε = 1/m, m N èqoume ìti f m = f 1/m f kaj c m. Mèqri t ra èqoume ìti f m f ston H l ). Gia na apodeðxoume to je rhma oloklhrwtikˆ prèpei na broôme mða akoloujða F m ston q ro C c ) ètsi ste F m f isqurˆ ston H l ). Gia na to petôqoume autì jewroôme kˆpoia sunˆrthsh k : [0, 1] ston q ro C c ) me kx) = 1 gia x 1. OrÐzoume thn akoloujða g m x) = kx/m)fx). Tìte g m H l ) m N, kai epiplèon èqei sumpagèc st rigma. Prˆgmati g m eðnai ston H l ) afoô f eðnai ston H l ) kai afoô k eðnai fragmènh me fragmènec parag gouc diìti suppd α k suppk kai k C c ). H sunˆrthsh g m èqei sumpagèc st rigma diìti k C c ). Epiplèon, f g m L ) = = x fx) k m) x m fx) dx fx) 1 k x m) dx

54 54 KEFŸALAIO 3. OI QŸWROI H S R N ) KAI H S,P R N ) + x >m x >m fx) 1 k x m) dx fx) dx 0 diìti 0 kx) < 1 kai kx) = 1 ìtan x 1. 'Eqoume dhlad ìti f g m L ) 0 kaj c m. Gia kˆje α = α 1 α...α n ) me α l D α f D α g m L ) = x D α fx) D k fx)) m) α dx = = x ) D α fx) fx)d α k k R m n x D β fx)d γ k m) dx β+γ=α β+γ=α x m) D α fx) x x 1 k D R m)) α fx) fx)d α k m) n x D β fx)d γ k m) dx x ) 4 1 k D α fx) dx R m n +4 fx) x ) D α k dx R m n +4 x ) D β fx)d γ k dx m β+γ=α 4 D α fx) C 1 dx + dx x m R m α fx) n C +4 fx) dx 0 m γ Dβ kaj c m. β+γ=α